3.566 \(\int \frac{\sqrt{d+i c d x} (a+b \sinh ^{-1}(c x))}{(f-i c f x)^{5/2}} \, dx\)

Optimal. Leaf size=185 \[ -\frac{i d^3 (1+i c x)^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 i b d^3 \left (c^2 x^2+1\right )^{5/2}}{3 c (c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{b d^3 \left (c^2 x^2+1\right )^{5/2} \log (c x+i)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]

[Out]

(((2*I)/3)*b*d^3*(1 + c^2*x^2)^(5/2))/(c*(I + c*x)*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - ((I/3)*d^3*(1 +
I*c*x)^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (b*d^3*(1 + c^2*x^2
)^(5/2)*Log[I + c*x])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))

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Rubi [A]  time = 0.306474, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {5712, 651, 5819, 12, 627, 43} \[ -\frac{i d^3 (1+i c x)^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 i b d^3 \left (c^2 x^2+1\right )^{5/2}}{3 c (c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{b d^3 \left (c^2 x^2+1\right )^{5/2} \log (c x+i)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[d + I*c*d*x]*(a + b*ArcSinh[c*x]))/(f - I*c*f*x)^(5/2),x]

[Out]

(((2*I)/3)*b*d^3*(1 + c^2*x^2)^(5/2))/(c*(I + c*x)*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - ((I/3)*d^3*(1 +
I*c*x)^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (b*d^3*(1 + c^2*x^2
)^(5/2)*Log[I + c*x])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))

Rule 5712

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :>
Dist[((d + e*x)^q*(f + g*x)^q)/(1 + c^2*x^2)^q, Int[(d + e*x)^(p - q)*(1 + c^2*x^2)^q*(a + b*ArcSinh[c*x])^n,
x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 + e^2, 0] && HalfIntegerQ[p,
q] && GeQ[p - q, 0]

Rule 651

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))
/(2*c*d*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rule 5819

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Wit
h[{u = IntHide[(f + g*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[Dist[1/Sqrt[1 +
c^2*x^2], u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[p + 1/2, 0]
&& GtQ[d, 0] && (LtQ[m, -2*p - 1] || GtQ[m, 3])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{d+i c d x} \left (a+b \sinh ^{-1}(c x)\right )}{(f-i c f x)^{5/2}} \, dx &=\frac{\left (1+c^2 x^2\right )^{5/2} \int \frac{(d+i c d x)^3 \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=-\frac{i d^3 (1+i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{\left (b c \left (1+c^2 x^2\right )^{5/2}\right ) \int -\frac{i d^3 (1+i c x)^3}{3 c \left (1+c^2 x^2\right )^2} \, dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=-\frac{i d^3 (1+i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{\left (i b d^3 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac{(1+i c x)^3}{\left (1+c^2 x^2\right )^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=-\frac{i d^3 (1+i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{\left (i b d^3 \left (1+c^2 x^2\right )^{5/2}\right ) \int \frac{1+i c x}{(1-i c x)^2} \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=-\frac{i d^3 (1+i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{\left (i b d^3 \left (1+c^2 x^2\right )^{5/2}\right ) \int \left (-\frac{2}{(i+c x)^2}-\frac{i}{i+c x}\right ) \, dx}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ &=\frac{2 i b d^3 \left (1+c^2 x^2\right )^{5/2}}{3 c (i+c x) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{i d^3 (1+i c x)^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{b d^3 \left (1+c^2 x^2\right )^{5/2} \log (i+c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.418799, size = 131, normalized size = 0.71 \[ -\frac{i d \sqrt{f-i c f x} \left ((c x-i) \left (a c x-i a+b \sqrt{c^2 x^2+1}\right )-b (c x+i) \sqrt{c^2 x^2+1} \log (d (-1+i c x))+b (c x-i)^2 \sinh ^{-1}(c x)\right )}{3 c f^3 (c x+i)^2 \sqrt{d+i c d x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[d + I*c*d*x]*(a + b*ArcSinh[c*x]))/(f - I*c*f*x)^(5/2),x]

[Out]

((-I/3)*d*Sqrt[f - I*c*f*x]*((-I + c*x)*((-I)*a + a*c*x + b*Sqrt[1 + c^2*x^2]) + b*(-I + c*x)^2*ArcSinh[c*x] -
b*(I + c*x)*Sqrt[1 + c^2*x^2]*Log[d*(-1 + I*c*x)]))/(c*f^3*(I + c*x)^2*Sqrt[d + I*c*d*x])

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Maple [F]  time = 0.313, size = 0, normalized size = 0. \begin{align*} \int{(a+b{\it Arcsinh} \left ( cx \right ) )\sqrt{d+icdx} \left ( f-icfx \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(c*x))*(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(5/2),x)

[Out]

int((a+b*arcsinh(c*x))*(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(5/2),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))*(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.73191, size = 1330, normalized size = 7.19 \begin{align*} -\frac{24 \, \sqrt{c^{2} x^{2} + 1} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} b c x + 12 \,{\left (b c^{2} x^{2} - 2 i \, b c x - b\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - 2 \,{\left (3 \, c^{4} f^{3} x^{3} + 3 i \, c^{3} f^{3} x^{2} + 3 \, c^{2} f^{3} x + 3 i \, c f^{3}\right )} \sqrt{\frac{b^{2} d}{c^{2} f^{5}}} \log \left (\frac{3 \,{\left (2 i \, b c^{6} x^{2} - 4 \, b c^{5} x - 4 i \, b c^{4}\right )} \sqrt{c^{2} x^{2} + 1} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} + 2 \,{\left (3 i \, c^{9} f^{3} x^{4} - 6 \, c^{8} f^{3} x^{3} + 3 i \, c^{7} f^{3} x^{2} - 6 \, c^{6} f^{3} x\right )} \sqrt{\frac{b^{2} d}{c^{2} f^{5}}}}{3 \,{\left (16 \, b c^{3} x^{3} + 16 i \, b c^{2} x^{2} + 16 \, b c x + 16 i \, b\right )}}\right ) + 2 \,{\left (3 \, c^{4} f^{3} x^{3} + 3 i \, c^{3} f^{3} x^{2} + 3 \, c^{2} f^{3} x + 3 i \, c f^{3}\right )} \sqrt{\frac{b^{2} d}{c^{2} f^{5}}} \log \left (\frac{3 \,{\left (2 i \, b c^{6} x^{2} - 4 \, b c^{5} x - 4 i \, b c^{4}\right )} \sqrt{c^{2} x^{2} + 1} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} + 2 \,{\left (-3 i \, c^{9} f^{3} x^{4} + 6 \, c^{8} f^{3} x^{3} - 3 i \, c^{7} f^{3} x^{2} + 6 \, c^{6} f^{3} x\right )} \sqrt{\frac{b^{2} d}{c^{2} f^{5}}}}{3 \,{\left (16 \, b c^{3} x^{3} + 16 i \, b c^{2} x^{2} + 16 \, b c x + 16 i \, b\right )}}\right ) + 3 \,{\left (4 \, a c^{2} x^{2} - 8 i \, a c x - 4 \, a\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f}}{3 \,{\left (12 \, c^{4} f^{3} x^{3} + 12 i \, c^{3} f^{3} x^{2} + 12 \, c^{2} f^{3} x + 12 i \, c f^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))*(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(24*sqrt(c^2*x^2 + 1)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*b*c*x + 12*(b*c^2*x^2 - 2*I*b*c*x - b)*sqrt(I*
c*d*x + d)*sqrt(-I*c*f*x + f)*log(c*x + sqrt(c^2*x^2 + 1)) - 2*(3*c^4*f^3*x^3 + 3*I*c^3*f^3*x^2 + 3*c^2*f^3*x
+ 3*I*c*f^3)*sqrt(b^2*d/(c^2*f^5))*log(1/3*(3*(2*I*b*c^6*x^2 - 4*b*c^5*x - 4*I*b*c^4)*sqrt(c^2*x^2 + 1)*sqrt(I
*c*d*x + d)*sqrt(-I*c*f*x + f) + 2*(3*I*c^9*f^3*x^4 - 6*c^8*f^3*x^3 + 3*I*c^7*f^3*x^2 - 6*c^6*f^3*x)*sqrt(b^2*
d/(c^2*f^5)))/(16*b*c^3*x^3 + 16*I*b*c^2*x^2 + 16*b*c*x + 16*I*b)) + 2*(3*c^4*f^3*x^3 + 3*I*c^3*f^3*x^2 + 3*c^
2*f^3*x + 3*I*c*f^3)*sqrt(b^2*d/(c^2*f^5))*log(1/3*(3*(2*I*b*c^6*x^2 - 4*b*c^5*x - 4*I*b*c^4)*sqrt(c^2*x^2 + 1
)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f) + 2*(-3*I*c^9*f^3*x^4 + 6*c^8*f^3*x^3 - 3*I*c^7*f^3*x^2 + 6*c^6*f^3*x)*
sqrt(b^2*d/(c^2*f^5)))/(16*b*c^3*x^3 + 16*I*b*c^2*x^2 + 16*b*c*x + 16*I*b)) + 3*(4*a*c^2*x^2 - 8*I*a*c*x - 4*a
)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f))/(12*c^4*f^3*x^3 + 12*I*c^3*f^3*x^2 + 12*c^2*f^3*x + 12*I*c*f^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asinh(c*x))*(d+I*c*d*x)**(1/2)/(f-I*c*f*x)**(5/2),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))*(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(5/2),x, algorithm="giac")

[Out]

Exception raised: AttributeError